Sequential Monte Carlo Techniques for Solving Non-Linear Systems

Given a system of m equations F(x) = 0 (where m is large and x is an unknown m-vector), we seek to apply sequential Monte Carlo [SMC] methods to find solutions efficiently. This paper follows up on a previous paper by the same author, in which consideration was limited to linear systems of the form Ax = a (where, again, m is large, A is a known (mxm) matrix, a is a known m-vector, and x is an unknown m-vector). It was shown there that effective techniques could reduce computation times dramatically (speed-up factors of 550 to 26,000 were obtained in sample calculations). The methods presented here rely on the use of Newtonian linearization, combined with the SMC methods previously described. Incidentally, the optimization of these SMC methods is discussed here and should clarify the parametrization of the SMC techniques so as to yield the highest efficiency.